
theorem Th21:
  for T,S being TopSpace st the carrier of T = the carrier of S
for R being Refinement of T,S for V being Subset of T, W being Subset of R st W
  = V holds V is closed implies W is closed
proof
  let T,S be TopSpace such that
A1: the carrier of T = the carrier of S;
  let R be Refinement of T,S;
  let V be Subset of T, W be Subset of R;
  assume
A2: W = V;
  assume V is closed;
  then
A3: V` is open;
  the carrier of R = (the carrier of T) \/ the carrier of S by YELLOW_9:def 6
    .= the carrier of T by A1;
  then W` in the topology of T by A3,A2;
  then W` is open by Th19;
  hence thesis;
end;
